3.145 \(\int (a+b \sinh ^3(c+d x)) \, dx\)

Optimal. Leaf size=32 \[ a x+\frac{b \cosh ^3(c+d x)}{3 d}-\frac{b \cosh (c+d x)}{d} \]

[Out]

a*x - (b*Cosh[c + d*x])/d + (b*Cosh[c + d*x]^3)/(3*d)

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Rubi [A]  time = 0.0211492, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2633} \[ a x+\frac{b \cosh ^3(c+d x)}{3 d}-\frac{b \cosh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Sinh[c + d*x]^3,x]

[Out]

a*x - (b*Cosh[c + d*x])/d + (b*Cosh[c + d*x]^3)/(3*d)

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \left (a+b \sinh ^3(c+d x)\right ) \, dx &=a x+b \int \sinh ^3(c+d x) \, dx\\ &=a x-\frac{b \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=a x-\frac{b \cosh (c+d x)}{d}+\frac{b \cosh ^3(c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0105271, size = 34, normalized size = 1.06 \[ a x-\frac{3 b \cosh (c+d x)}{4 d}+\frac{b \cosh (3 (c+d x))}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sinh[c + d*x]^3,x]

[Out]

a*x - (3*b*Cosh[c + d*x])/(4*d) + (b*Cosh[3*(c + d*x)])/(12*d)

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Maple [A]  time = 0.005, size = 28, normalized size = 0.9 \begin{align*} ax+{\frac{b\cosh \left ( dx+c \right ) }{d} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*sinh(d*x+c)^3,x)

[Out]

a*x+b/d*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c)

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Maxima [A]  time = 1.15456, size = 80, normalized size = 2.5 \begin{align*} a x + \frac{1}{24} \, b{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} + \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sinh(d*x+c)^3,x, algorithm="maxima")

[Out]

a*x + 1/24*b*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d)

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Fricas [A]  time = 1.90505, size = 128, normalized size = 4. \begin{align*} \frac{b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 12 \, a d x - 9 \, b \cosh \left (d x + c\right )}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sinh(d*x+c)^3,x, algorithm="fricas")

[Out]

1/12*(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + 12*a*d*x - 9*b*cosh(d*x + c))/d

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Sympy [A]  time = 0.549777, size = 41, normalized size = 1.28 \begin{align*} a x + b \left (\begin{cases} \frac{\sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 \cosh ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \sinh ^{3}{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sinh(d*x+c)**3,x)

[Out]

a*x + b*Piecewise((sinh(c + d*x)**2*cosh(c + d*x)/d - 2*cosh(c + d*x)**3/(3*d), Ne(d, 0)), (x*sinh(c)**3, True
))

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Giac [A]  time = 1.12415, size = 72, normalized size = 2.25 \begin{align*} a x - \frac{{\left ({\left (9 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-3 \, d x - 3 \, c\right )} - e^{\left (3 \, d x + 3 \, c\right )} + 9 \, e^{\left (d x + c\right )}\right )} b}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sinh(d*x+c)^3,x, algorithm="giac")

[Out]

a*x - 1/24*((9*e^(2*d*x + 2*c) - 1)*e^(-3*d*x - 3*c) - e^(3*d*x + 3*c) + 9*e^(d*x + c))*b/d